High-dimensional Black-box Optimization via Divide and Approximate Conquer

TL;DR

This paper introduces Divide and Approximate Conquer (DAC), a method that efficiently solves high-dimensional black-box optimization problems with interdependent sub-problems by approximating partial solutions, reducing computational costs while ensuring convergence.

Contribution

It proposes a novel approximation approach for divide and conquer in high-dimensional optimization, significantly reducing evaluation costs and guaranteeing convergence.

Findings

DAC reduces evaluation time from exponential to polynomial.

Empirical results show DAC effectively solves non-separable high-dimensional problems.

DAC maintains convergence to the global optimum despite approximation.

Abstract

Divide and Conquer (DC) is conceptually well suited to high-dimensional optimization by decomposing a problem into multiple small-scale sub-problems. However, appealing performance can be seldom observed when the sub-problems are interdependent. This paper suggests that the major difficulty of tackling interdependent sub-problems lies in the precise evaluation of a partial solution (to a sub-problem), which can be overwhelmingly costly and thus makes sub-problems non-trivial to conquer. Thus, we propose an approximation approach, named Divide and Approximate Conquer (DAC), which reduces the cost of partial solution evaluation from exponential time to polynomial time. Meanwhile, the convergence to the global optimum (of the original problem) is still guaranteed. The effectiveness of DAC is demonstrated empirically on two sets of non-separable high-dimensional problems.